Theorem REF  is a special case of a much more general result, which involves piecewise monotone maps. To give a flavor of this more general result we will formulate now its special case which involves not only Gauss transformation \(T_G\)  and times \(b\)  transformations \(T_b\) , but also \(\beta \) -transformations \(T_{\beta }\) . Recall that for \(\beta >1\) , \(\beta \notin \mathbb {N}\) ,
\(\beta \) -transformation \(T_{\beta }\)  is given by \(T_{\beta } x = \beta x \bmod \, 1\) . These transformations were introduced by Rényi ([1]}) who in particular showed that there exists a unique \(T_{\beta }\) -ergodic probability measure \(\mu _{\beta }\)  which satisfies \(1 - \frac{1}{\beta } \le \frac{d \mu _{\beta }}{d \lambda } \le \frac{1}{1 -1 / \beta }\) .
